Optimal. Leaf size=54 \[ \frac{i b \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{\log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d} \]
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Rubi [A] time = 0.0711642, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4868, 2447} \[ \frac{i b \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{\log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)} \, dx &=\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{(b c) \int \frac{\log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{d}+\frac{i b \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0556677, size = 102, normalized size = 1.89 \[ \frac{i b \text{PolyLog}(2,-i c x)}{2 d}-\frac{i b \text{PolyLog}(2,i c x)}{2 d}+\frac{i b \text{PolyLog}\left (2,-\frac{c x+i}{-c x+i}\right )}{2 d}+\frac{\log \left (\frac{2 i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}+\frac{a \log (x)}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.051, size = 193, normalized size = 3.6 \begin{align*} -{\frac{a\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,d}}-{\frac{ia\arctan \left ( cx \right ) }{d}}+{\frac{a\ln \left ( cx \right ) }{d}}-{\frac{b\arctan \left ( cx \right ) \ln \left ( cx-i \right ) }{d}}+{\frac{b\arctan \left ( cx \right ) \ln \left ( cx \right ) }{d}}+{\frac{{\frac{i}{2}}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) }{d}}-{\frac{{\frac{i}{2}}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) }{d}}+{\frac{{\frac{i}{2}}b{\it dilog} \left ( 1+icx \right ) }{d}}-{\frac{{\frac{i}{2}}b{\it dilog} \left ( 1-icx \right ) }{d}}+{\frac{{\frac{i}{2}}b\ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) \ln \left ( cx-i \right ) }{d}}+{\frac{{\frac{i}{2}}b{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{d}}-{\frac{{\frac{i}{4}}b \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, b{\left (\frac{i \, \arctan \left (c x\right )^{2}}{d} - 2 \, \int \frac{\arctan \left (c x\right )}{c^{2} d x^{3} + d x}\,{d x}\right )} - a{\left (\frac{\log \left (i \, c x + 1\right )}{d} - \frac{\log \left (x\right )}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59776, size = 109, normalized size = 2.02 \begin{align*} \frac{-i \, b{\rm Li}_2\left (\frac{c x + i}{c x - i} + 1\right ) + 2 \, a \log \left (x\right ) - 2 \, a \log \left (\frac{c x - i}{c}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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